Towards the Jantzen conjecture. II

C ^OMPOSITIO M ^ATHEMATICA

A. J ^OSEPH

Towards the Jantzen conjecture. II

Compositio Mathematica, tome 40, n

^o

1 (1980), p. 69-78

<http://www.numdam.org/item?id=CM_1980__40_1_69_0>

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69

TOWARDS THE JANTZEN CONJECTURE II

A.

Joseph*

COMPOSITIO MATHEMATICA, ^Vol. 40, ^Fasc. 1, 1980, pag. 69-78

(c) ¹⁹⁸⁰ Sijthoff &#x26; Noordhoff International Publishers - Alphen ^{aan den} Rijn

Printed in the Netherlands

Abstract

A

recently developed additivity principle

^{for Goldie rank is com-}

bined with the results of the first _paper of this series to establish a

very

slightly

weaker form of the Jantzen

conjecture

^{for the}

primitive

spectrum ^{of a}

complex simple

^Lie

algebra

^of type

An.

^{Precise in-} formation on the

primitive

spectra ^{in other}

simple

^Lie

algebras

^{and on}

the Goldie ranks of the associated

quotient algebras

^{is also obtained.}

1. Introduction

Let q

^{be a}

complex semisimple

^Lie

algebra, U(g)

^its

enveloping algebra

^{and Prim}

U(g)

^{the set of}

primitive

ideals of

U(g).

^The

classification of Prim

U(g) for A simple

^of type

An-t (Cartan notation)

for n _ 6

(and

^{several other low rank}

cases)

^was

given by

^{Borho and}

Jantzen

[2], [3],

^and from their results Jantzen

[1],

^5.9

guessed

^its

solution for

general

^{n. In}

[8], 8.2, 11.8,

^we

suggested

^{what form this}

conjecture

should take for an

arbitrary semisimple

^Lie

algebra

^and

indicated how this should be related to the Goldie ranks of the

quotient algebras U (g)/ 1:

^{I E Prim}

U(g).

^{In the} present ^{work we} show how the results of

[8]

combined with a

specially developed additivity principle

for Goldie rank

[9]

^{lead to a}

proof

^{of a} very

slightly

weaker form of Jantzen’s

conjecture

^{and also to some}

quite precise

^{results in the}

general

^{case. A further result} which obtains from 5.3 and

[8],

^7.7 is that for

regular

central characters the anni-

* This paper ^was written while the author was a guest of the Institute for Advanced Studies, ^Hebrew University, Jerusalem and on leave of absence from the Centre Nationale de la Recherche Scientifique.

0010-437X/80/01/0069-10 $00.20/0

hilators of the

simple subquotients

^{of a}

given principal

series module for

SL(n,

C) ^are

pairwise disjoint.

^The notation is that of

[8].

The contents of this _paper were

reported

^{at a}

meeting

^{on envelo-}

ping algebras

^at Oberwolfach

during

^5-11

February

^1978.

2.

Multiplicities

2.1 Fix a

triangular decomposition g = n (9 b (D n-

^for g, ^set b:= ⁿ

Et&#x3E; b

^{and for each A Et)}

^*,

^let

M(A):= U(g)@UbCA-p

^{denote the}

associated Verma module

[8],

^{3.1. Given M a}

finitely generated

^left

U(,g)

^{module define the} Gelfand-Kirillov dimension

d(M) (resp.

multiplicity e(M))

^{of M as in}

[8],

^2.1. Given left

U(g) modules, M,

^N define

Homc(M, N)

^{as a}

left U : = U(g) Q U(g)

^module

([8], 3.2)

^and

let

L(M,

N) denote the U submodule of

ad ()

^{finite elements.}

(We

recall in

particular

that each X E n - defines a

locally nilpotent

^deriva-

tion ad X of

L(M, N».

^{Set n = dim n - and let r be} the smallest

positive integer

^{such that}

(ad n-)r g

^{= 0.}

LEMMA: Fix A, JL ^E b* ^{and let M}

(resp. N)

^{be a}

subquotient of M(A ) (resp. M(li». Suppose d(L(M, N)) &#x3E; d(M)

⁺

d(N).

^Then

equal- ity

^{holds and}

This is _very modest

generalization

^of

[8],

^{6.4 and for which the}

proof

^needs

only

^the

following

^trivial

changes. Replace

^{m -}

by

^{n - and}

VB{Á) (resp. VB’(JL» by any b

^stable finite dimensional

generating subspace M° (resp. N)

^{of M}

(resp. N).

2.2 Relative to the

given triangular decomposition

of g, ^{let R}

(resp.

R+)

denote the set of all non-zero

(resp. positive)

^{roots and B C R+}

the set of

simple

^{roots. Let s denote the}

largest

coefficient of a

simple

root that can occur in a

positive

^{root and set t = sn : n =} dim n -. Given V ^a

locally finite b module,

^let

V ,: li ^{e $ *}

denote its

weight subspace

of

weight

1£.

LEMMA: Let M be a

simple subquotient of

^{a Verma} module and set I ⁼ Ann M. Then

The first assertion is

just [7],

2.7 and the second a refinement of it.

71

Let T

(resp. T-, E)

^{denote the}

image

^of

n (f) C (resp. n’@C, g e C)

ⁱⁿ

U(n)/(I

^fl

U(n)) (resp. U(n-)/(I

^f1

U(n-)), U(g)/I).

^{Let e denote the}

canonical generator of ^M which we suppose has

weight

^À and for all k E N ^set

M"‘ : = T ke = E ke.

Define an

ordering a

^{on ZB}

through

IL &#x3E;

given

IL - v E NB. For

each k E N, Mk

^{is a} finite direct sum of its

weight subspaces (Mk)A-v

^of

weight À - v: v E ZB

^{and we set}

Qk

⁼

f v

^{E Z B :}

(Mk)A-v;;é 01.

^{Choose a maximal}

element IL

^E

f2k-

^Since

M is a

simple U(g)

^{module there} exists ^for each ^m E

(Mk)A-p.

^an

element a E

U(g)

^{such that am = e. From the}

decomposition U (g) = U(n-) @ U(n)@ U(h)

and the fact that

MO

is a one dimensional

highest weight

space for M of

weight

^{À we can} suppose that a E

U(n),,.

^{Then for all V E}

nk B{IL}

^{we have}

a(Mk)A-v

^C

MA+p.-v

^{= 0 where}

the last step ^{holds because} IL - v E N B. Now Mk is b stable so

by

induction on &#x3E; it follows

easily

^that

Hom(Mk, MO)

^C

© ^U(n)vIMk.

vEllk

Conversely

^{from the relation}

Mk

⁼

T k e,

^{we obtain}

Hom(M°, Mk) C Tk 1 mo

^and

For each v E N B, a E B, ^let r§ ^{dénote the} coefficient of a in v.

Recalling

^the définition

of t,

it follows for each v

E !Jk

^that

On the other hand if a E

S(n)

^is

homogeneous

^of

degree

^{m and of}

weight v,

^then

Hence

Hom(Mk ,M)C TktlMk

^{and so}

Hom(Mk, Mk) C T kT k, lfk *C Eko+t)IMk,

^which

gives

Yet

by

^definition

Equating

powers of k

gives (ii).

2.3 Let M be the

subquotient

^{of a Verma module.}

By [2], 3.16,

there exists a

positive integer

^u

depending only

^on g such that the

length

^{of M is less than u. Define} n, r, t ^as in 2.2.

PROPOSITION: Let M be a

subquotient of

^{a Verma module. Set}

I ⁼ Ann M. Then

(i) d( U(4)lI)

⁼

d(L(M, M»

^{= 2}

d(M).

(ii) e(L(M, M)) S (4rtu )2n e( U(g)/I).

Since

L(M, M)

^contains

U(g)/I

^{as a}

U(g) submodule,

^{we obtain}

(i)

from 2.2

(i)

^{and the first} part ^of 2.1. From the definition of u and _say

[8],

^{2.2 we can choose a}

simple subquotient

^{L of M}

satisfying d(L)

⁼

d(M)

^{and u}

d(L) &#x3E; d(M).

^{Since I C Ann L, we obtain}

(ii)

^from

2.1 and 2.2

(ii).

3. Goldie Rank

3.1 Let B be a

prime (left

^and

right)

Noetherian

ring.

^{Recall that}

the Goldie rank rk B is

just

^{the maximum number of direct} summands of left (or

right)

^{ideals of B. Given K a} left ideal of B let rk K denote the maximum number of direct summands of left ideals of B in K.

Given e an

idempotent

^{of B we remark that eBe is a}

prime

Noetherian

subring

^{of B and rk eBe = rk Be.}

3.2 Let AB ^be

simple

^Artinian

rings

^{with H a}

subring

^{of 0,8 and}

containing

^the

identity

^{1 of lB. Then z : :=}

(rk lB)1(rk A)

^E N+ (see

[5], Prop.

2, p.

137).

Given b E

lB,

^set

e(b)

⁼

f a

^{E A ab =}

01.

^The

following

is a

straightforward

exercise in

idempotent manipulations using [5],

Thm. 2, p. 47 and

Prop. 4,

p. 51.

LEMMA: There exists a set

(h i,

h2, ^...,

hz) of pairwise orthogonal idempotents in B satisfying t(hi)

^{= 0}

for

^{all i = 1,}

2,...,

^z.

3.3 Let A, B be

prime

Noetherian

rings

^{with A a}

subring

^{of B.}

Assume that the set S of

regular

^{elements of A is a subset of the}

regular

elements of B and is an Ore subset for both A and B and such that

S-IA

and

S-’B

are

simple,

^Artinian.

Suppose

further that A and B are

finitely generated

^left

U(g)

^modules.

COROLLARY:

If d(A)

⁼

d(B),

^then

Apply

3.2 with A ⁼

S-’A, e

⁼

^S-’B.

Choose s E S such that

h;s

^E

B for all i. Then

Ah1 + Ah2 + ... + Ahz

^{is a direct sum of}

U(g)

submodules of B each

isomorphic

^{to A.}

By

say

[8],

^{2.2 this}

gives

^the

required

^assertion.

73

4. The

additivity principle

4.1 Let F be a commutative

field, U(a)

^the

enveloping algebra

^{of a}

finite dimensional F-Lie

algebra

^{a. Let A be a}

quotient

^of

U(a)

^and

assume that A embeds in a

prime ring

^{B with}

identity

^{1 E A. Assume}

that B is

finitely generated

^{as a left and a}

right U(a)

module and that each b E B is

locally

^{ad a finite}

(c.f. [8], 2.3).

^Let

{PI,

P2, ...,

Pr}

^{be the}

set of minimal

primes

^{of A.}

By [9],

^{3.9 there exist}

zf E

^{N+ such that}

Here we obtain further information ^on the

zA.

4.2 Let S be the set of

regular

^{elements of A.}

By [9], 3.5, S

^is contained in the set of

regular

^{elements of B and}

by [9],

^{3.7 it is an}

Ore subset in both A and B and sî:=

S-’A

is Artinian and 00 :=

S-’B simple

Artinian. Given C a

subring

^{of e and T a subset of 00 we set}

tc(T) = {c E C: eT = O}, rc(T) = {c E C: Tc = O}.

^{When C = A we}

drop

^the

subscript.

Given P a minimal

prime

^of

A,

then P n S

= 0

and so

S-’P

⁼

PS-1

is a

(minimal) prime

^{of A}

(see

^for

example [10], 2.5-2.10).

^Set

BI = tB(P), B2

⁼

rB(BI),

B3 ⁼

fb

^{E B:}

Blb

^C

BI}.

^{These are all}

U(a)@ U(a)

submodules of B. Also B2B ^C B2, B 1 B2 ⁼ 0, B3::&#x3E; BI

and

so

Bil(Bi

ⁿ

B2), B31B,

have natural

F-algebra

structures.

Again

BBI ^C BI ^{and so}

S-IBI

^{is a} left ideal of the

simple

^Artinian

ring 00

^{and so we}

may write

S-IBI =

^{9Be for some}

idempotent e

^{E 00.}

Again

B3 :) ^{A and} A n Bi

=,e(P)

^{and so}

A/t(P)

^{embeds in}

B3/BI.

LEMMA:

(iv) Bii(Bi

ⁿ

B2) (resp. B31BI)

^{is a}

prime

Noetherian

ring

^and

Fract

Bil(Bi

^fl

B2) (resp. B3/BI)

^is

naturally isomorphic

^{to eee}

(resp.

( 1- e)@(I- e)).

^In

particular

^rk

B1/(BI

ⁿ

B2)

^{+ rk}

B3lBl

^{= rk B.}

(ii)

^{is clear}

for j

^{= 1.}

Again S-IB1

⁼

S-ItB(p) = lB(P) = tOO(PS-I) = tOO(S-lp).

^This

gives (i)

^{and the inclusion}

S-IBt:) B,S-1. Conversely given s E S,

^{b E}

Bi

we ^can

choose t E S,

^{c E B such that}

s-lb

⁼

ct -’ .

Then

ct-’P

⁼

s-’bP

⁼ 0 and so c E B,. This establishes

(iii) for j

^{= 1.}

Similarly (using (ii) for j

⁼

1)

^{we obtain}

S-1B2

⁼

B2S-1= (1 - e)@

^and

S-IB3

⁼

B3S-1 =

^{ee +}

( 1 - e)9B( 1 - e).

^This

gives (ii)

^and

(iii).

^Given

x E

S-IB1 n S-IB2,

^then

by

^the Ore condition there exists s E S such

that sx E B n

S-’B,

^fl

S-’B2

⁼ B1 ⁿ B2

by (ii).

^{A similar}

computation

on the

right

^and

(iii) gives S-I(BI

ⁿ

B2)

⁼

S-’Bi

^fl

S-lB2

⁼

(1- e)9Be

=

Bis-,

ⁿ

B2S-1 = (B,

ⁿ

B2)S-’.

Given s E S,

b + (BI n B2) E BI/(BI n B2) (resp. b + BI EE B31BI)

such that sb E Bi ^fl B2

(resp.

^{sb E}

BI)

^{then b E}

S-IBI

^rl

S-IB2

^{n B =}

B,

^fl

B2 (resp.

^{b E}

S-1B,

^{fl B =}

BI).

^A similar result holds for

right multiplication

^and combined with our

previous

observations establishes

(iv).

4.3 Retain the notation and

hypotheses

^of 4.1 and 4.2. Take an

ordering

of the minimal

primes

^{of A so that}

e(PI) 0

⁰

(this

^is

possible by

say

[9], 2.4)

^{and take P =} PI in 4.2. Recall 4.2

(iv).

PROPOSITION: :

Let N denote the nilradical of A and recall

([9], 2.7)

that .N’ : ⁼

S-I N

is the nilradical of A.

Again (c.f. [9], 3.9) (Q;

⁼

5"’P,:

^{i =}

1,2,..., rl

is the set of

primes

^{of the Artinian}

ring

^{A and Fract}

A/Pi = Al Qi,

up to

isomorphism.

^{Further recall}

([9], 3.8)

that there exists a set

{eh

^{e2, ...,}

erl

^of

pairwise orthogonal idempotents

^{for A}

satisfying X ei

^{= 1 and such that}

Qi = ( 1- ei)si

^{+ .N’.}

Taking i

^{= 1 and}

applying

^4.2

(i)

^{we obtain e =} eel. This

gives

^{eei = eele; = 0, for all i =}

2, 3,...,

^r.

Now recall that

Ale(Pi)

identifies as a

subalgebra

^of

B31BI. By

^4.2

(iv)

^we may

apply [9],

^3.8

(i)

^{to both}

Ale(PI)

^{and A. This}

gives

^for

i&#x3E; 1

This

gives (ii). (i)

follows from

(11),

^4.2

(iv)

^and

[9],

^3.8

(11).

4.4 Retain the notation and

hypothèses

^of 4.3. Let M be a faithful B module. Recall

(c.f. [9], 2.4)

^{that some}

product

^{of the}

Pi

^must vanish and let

L(A)

^{be the}

length

^{of a shortest}

product.

^{Given P a}

minimal

prime

of A such that

é(P) # 0,

^then

iE(A/é(P)) f(A).

Consider M as a

U(a)

module and B as a

U(a) 0 U(a)

^module.

THEOREM: M admits a normal series M ⁼

Aft D M2 D ... D ^{Mi+i =}

0 such that

for all

s = 1 , 2, ..., t ^{and all i =}

1, 2,...,

^r

(i)

^Ann

Ls,

^where

L, :

⁼

MIM,,,, is a

^minimal

prime of

^A.

75

(ii)

^{B admits a}

subquotient B(s) ^naturally isomorphic to

^a

prime

Noetherian

subring of HomF(Ls, Ls).

(iii)

Choose a minimal

prime

^{P of A such that}

t(P) 0

^{0 and set}

M2

⁼

{m

^{E M:}

B 1 m

⁼

0}.

^Then M2 D ^{PM and since M is a faithful B} module it follows that BI =

fb

^{E M :}

bM2

⁼

0}.

^Now

BIB2M

⁼ 0, ^so B2M ^C M2 ^{and since M is a faithful B module it} follows that

B2 = {b E B: bM c M2}.

^Hence

B(j): = B il(Bi n B2) (which

^{is a}

subquotient

^of

B)

identifies with a

subring

^of

HomF(L1, Ll). By

^4.2

(iv)

^it

is

prime

Noetherian.

Now

e(P)M2

^C BIM2 ^{= 0 and so}

e(P)

^Ann

LI

^{= 0. Since}

e(P) 0

⁰

by hypothesis,

^{it follows from}

[9],

^2.2

(ii),

^2.6

(i),

3.1, 3.6, ^that

d(A/Ann LI)

⁼

d(A)

⁼

d(A/P).

^Yet

Ann LI :)

^{P and so}

by [9],

^2.5

(i) equality

^{must hold.}

Again

^{since M is a faithful B}

module,

it follows that B3M2 C M2 ^{and so}

B31B, (which

^{is a}

subquotient

^of

B)

^identifies with a

subring

^of

HomF(M2, M2). By

^4.2

(iv)

^{it is}

prime

^Noetherian

and we recall that it contains

A/t(P)

^{as a}

subring.

^Hence

(i)

and

(ii)

obtain

by

^{induction on}

L(A)

^{and then}

(iii)

^{follows from 4.3.}

5. Main theorem

5.1 Let W be the group

generated by

^{the sa : a E R. Given B’ C} B,

set R’ = ZB’ n Rand define _pa E

S(b)

^{as in}

[8],

^6.6.

(Up

^{to a}

^scalar,

pB, is the

product

^{of the roots in R’ n R+ and 2}

^deg

^{pp = card}

^R’.)

^Let PB, ^{be the}

simple

^W submodule of

S(H)m generated by

pB,, with

m ⁼

deg

pB-

(c.f. [8], 8.6)

^{and let}

f2R (resp. n’R)

^denote the subset of

W

defined

by

^the P£t’: ^{B’ C B}

(resp.

P£t, ^C

S(b)m:

^{B’ C}

B).

5.2 Given À E b* ^define

BA, WA

^{as in}

[8],

3.3. Recall that Wx ^is

again

a

Weyl

group ^{so we} may define

corresponding

^subsets

f2R,À, n’R,A

^of

WA.

^Let

L(À)

^{denote the}

unique simple subquotient

^of

M(A)

^{and set}

7(A)

^{= Ann}

L(À). Set À

⁼

WÀ, Xî

⁼

{I (, ): p,

^E

Wk}

^{and for each m E} N, ^set

aeT={IEae).:d(U(g)/I)=cardR-2m}.

^{Call k}

regular

^if

(À, a ) # 0,

^{for all a E R.}

THEOREM: Assume À

EE regular

and that there exists ^{w E} W such that

wBA

^{C B. Then}

for

^{ail m E N one has}

By [6],

^{4.2 we can} assume BA ^{C B without loss of}

generality.

Then for each B’C B,, let M : =

MB,(A)

^{be the induced module defined}

by

B’, A ^{and set I =}

IB,(A) i=

^Ann

MB,(Á). By [8],

^4.3

(11), 4.8, 5.10,

^it follows that

L(M, M)

^{is a}

prime

Noetherian

ring

^of

locally ad g

^finite

elements,

^is

finitely generated

^{as a left and as a}

right U(g)

^{module and} rk

L(M, M)

⁼

pB,(A ).

^{Set m}

= deg

pB,. We

apply

^{4.4 with A =}

U(g)/I,

B ⁼

L(M, M).

^Let

fPI,

P2,...,

Pr)

denote the set of minimal

primes

^of

A. We have

Pi E f!£i

^and

by

^2.3

(i)

^and

[9],

^3.9

(i)

^that Pi ^E

Xm,

^{for all}

i.

By [9],

^3.9

(ii)

^{there exist} zi ^E N+ such that

By

^2.3

(ii), 3.3,

^4.4

(iii), [2],

3.16, ^{there exists a}

positive integer c(g) depending only

on g

(in particular independent

^of A) ^{such that}

zi

c(g).

^Then

by [8],

6.5, it follows that the assertion of the theorem holds for at least some À E

b *. By [2],

^{2.12 it must} then hold for all A

regular.

REMARKS : The

inequality

^{can be strict. For}

example take g simple

of type

D4

^{with m = 7}

(see [8], 11.7). Using [2], 2.14,

^{the technical}

assumption

wBA ^{C B can be} weakened: for

example

^{it is}

enough

^that

every strict subset of B, ^{can be}

conjugated

^{into B.}

Finally

^{for À}

non-regular

^recall

[2],

^2.12.

5.3 Assume that the

simple

^factors

of g are

^{all of} type An.

Let l’A

denote the set of involutions of

WA.

COROLLARY: For all À E t) ^*

regular

^{one has}

card

H

^{= card}

E,,.

It suffices to recall that

f2R,À = W"

^{in this case}

(c.f. [8], 8.4)

^{and to}

apply [4], Prop.

^{9 to 5.2.}

REMARKS : Note in

particular

^that this establishes

[8],

^{10.3. This}

gives

^{a natural}

partition

^of

H

^into

disjoint

^subsets

(H_):

^{u E}

Wa

satisfying card(H)u

^{= dim for À}

regular. Again

^{for À}

regular

^and

each I e

(Kg)

^{there exist}

exactly

dim a elements of

W,

^{such that}

I ⁼

I(wÀ): W

^E

W, (fixing

say -À

dominant).

^{Thus to establish the}

Jantzen

conjecture [1], 5.9,

it remains to show that for each I E

(H),

the zero

variety

^of gr I is the closure of the

appropriate

^Richardson

orbit

[8],

^{8.2. We remark that} this orbit can also be viewed as the

77

image

of o- under the

Springer

map (see

[8], 11.8).

^{For this it suffices}

to establish

[8],

^{9.1- that is to show that}

gr J

^is

primary

^for every induced ideal J. So far we have

only

^{been able to} show that the dimension of the zero

variety

^of

gr 1

^{has the}

appropriate

^value

[7],

4.1.

5.4 For type An the lower bound 5.2 on card

:JeT

^{is in fact its exact}

value. This is _very

nearly

^{true in}

general

^as may be seen

by examining

the upper bound

implied by [6],

^{5.1 and}

[7],

^{3.1. Even} aside from the combinatorial

questions involving

^the

Weyl

group, ^we do not yet ^have

enough

information to

give

^as

complete

^{a solution as in} type An. ^The main

difficulty

^{arises from the} presence of non-Richardson-like orbits.

Nevertheless we can extend the low rank

computations

^{of Borho and}

Jantzen

(c.f. [7],

Sect. 5 and

[8],

^Sect.

11.8)

^to type B4 (or C4). ^{In this}

case the _upper and lower bounds for card

ae:

^{-A E}

^P(R)++,

^coincide

and

give

^card

f

^{= 50: -A E}

P(R)++.

^In type D4 ^the upper and lower bounds also coincide for m values

corresponding

^to Richardson orbits and the total number of such ideals is 32. One further obtains card

ae1

^{= 2 or a, card}

X’

^{= 4: -A E}

P(R)",

where the latter

corresponds

^{to the} almost maximal ideals

(c.f. [2],

^{4.5 and}

[8], 11.8).

By [3]

the total number of ideals is 36.

Finally

^we remark that the upper bound on the zi in

(*) implies

^that

they

^{are constant in some} Zariski dense subset of

P(R)". Consequently

^{if the} upper and lower bounds coincide for a

given

^{value of m, then}

by (*)

^{the Goldie ranks} of the

corresponding quotient algebras

^are defined in this subset

by

^a basis of

@(P*,:

^{B’ C B : card R’ =}

2ml.

This establishes a weak ver-

sion of

[8],

^11.1.

REFERENCES

[1] W. BORHO: Recent advances in the enveloping algebras of semi-simple ^Lie algebras. ^Sem. Bourbaki, ^no. 489, Nov. 1976.

[2] ^{W. BORHO} and J.C. JANTZEN: Über _primitive Ideale in der Einhüllenden einer halbeinfacher Lie algebra. ^Invent. Math., 39 (1977) ^1-53.

[3] W. BORHO and J.C. JANTZEN: (unpublished).

[4] M. DUFLO: Sur la classification des idéaux primitifs ^dans l’algèbre enveloppante d’une algèbre ^{de Lie} semi-simple. ^Ann. Math., 105 (1977) pp. 107-130.

[5] N. JACOBSON: Structure of Rings. ^{Amer. Math. Soc.} Colloq. Publ., Vol. XXXVII, 1974.

[6] A. JOSEPH: A characteristic variety ^{for the} primitive spectrum ^{of a} semisimple

Lie algebra, (unpublished). Short version in LN 587 (1977) pp. 102-118.

[7] A. JOSEPH: Gelfand-Kirillov dimension for the annihilators of simple quotients ^of

Verma modules. J. Lond. Math. Soc., ¹⁸ (1978) ^{pp. 50-60.}

[8] A. JOSEPH: Towards the Jantzen conjecture. Compositio ^{Math. 40} (1980) ^35-67.

[9] A. JOSEPH and L.W. SMALL: An additivity principle ^{for Goldie} rank. Israel J.

Math. 31 (1978) ^105-114.

[10] W. BORHO, ^P. GABRIEL and R. RENTSCHLER: Primideale in Einhüllenden auflösbarer Lie-algebren, ^LN 357, Springer-Verlag, Berlin-Heidelberg-New ^York,

1973.

(Oblatum 2-IV-1978 &#x26; 28-XI-1978) ^{Centre de} mathématiques Bâtiment 425

91405 Orsay France